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LOW‐LYING ZEROS OF L ‐FUNCTIONS FOR MAASS FORMS OVER IMAGINARY QUADRATIC FIELDS
Author(s) -
Liu ShengChi,
Qi Zhi
Publication year - 2020
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/mtk.12041
Subject(s) - mathematics , infinity , eigenvalues and eigenvectors , pure mathematics , quadratic equation , laplace transform , mathematical analysis , geometry , quantum mechanics , physics
Abstract We study the 1‐ or 2‐level density of families of L ‐functions for Hecke–Maass forms over an imaginary quadratic field F . For test functions whose Fourier transform is supported in ( − 3 2 , 3 2 ) , we prove that the 1‐level density for Hecke–Maass forms over F of square‐free level q , as N ( q ) tends to infinity, agrees with that of the orthogonal random matrix ensembles. For Hecke–Maass forms over F of full level, we prove similar statements for the 1‐ and 2‐level densities, as the Laplace eigenvalues tend to infinity.